# Checking method for calculating free charge from Electric field in LIH

1. May 12, 2012

### tomwilliam

1. The problem statement, all variables and given/known data

I'm asked to find the free charge per unit length on a cylinder which is surrounded by a LIH dielectric material. I have an expression for the electrostatic potential $$V(r)$$ in cylindrical coordinates.

2. Relevant equations
$$\mathbf{D}=\varepsilon \varepsilon_0 \mathbf{E}$$
Gauss's Law in media equating flux of displacement field across a surface to the free charge.
$$\mathbf{E}=-grad V$$

3. The attempt at a solution
I've calculated E using the last equation there, then used it (and the LIH assumption) to calculate D, and now I can equate:
$$D_r \times 2 \pi r L = Q_f$$
Where L is the length of the cylinder and Q_f is the free charge.
This is enough to give me the expression for free charge per unit length. My problem is: am I right in taking the value of E at the cylinder surface?
I know that the potential is zero on this surface, so that seems to throw my working out a little.

2. May 12, 2012

### Gordianus

Why do you think the potential should be zero on the surface?

3. May 12, 2012

### tomwilliam

I know that the potential is zero at the surface because I have the expression for V(r) and it works out as zero on the surface.

I've just realised, though, that I could choose an arbitrary Gaussian surface in the dielectric and use that to calculate the free charge per unit length.

Would that work?